A $C^1$-continuous finite element formulation for solving the Jeffery-Hamel boundary value problem

TL;DR

This paper develops a C^1 finite element method for solving the Jeffery-Hamel boundary value problem, demonstrating nearly optimal convergence rates despite non-coercivity.

Contribution

It introduces a novel C^1 finite element formulation for the Jeffery-Hamel problem, highlighting its advantages and analyzing its convergence behavior.

Findings

Converges at nearly optimal rates (O(h^4)) in L^2 and H^1 norms.

The formulation is non-coercive, complicating theoretical analysis.

Numerical solutions with quartic Hermite elements are effective.

Abstract

The third-order Jeffery-Hamel ODE governing the flow of an incompressible fluid in a two-dimensional wedge is briefly derived, and a C^1 finite element formulation of the equation is developed. This formulation has several advantages, including a natural framework for enforcing the boundary conditions, a numerically efficient solution procedure, and suitability for implementation within well-established, open, scientific computing tools. The finite element formulation is shown to be non-coercive, and therefore not ideal for proving existence, uniqueness, or a priori error estimates, but the numerical solutions computed with quartic Hermite elements are nevertheless found to converge to reference solutions at nearly optimal rates (O(h^4) in both L^2 and H^1 norms). Further work is required to better understand the cause of the suboptimal convergence rates, and a linear model problem which exhibits analogous characteristics is also discussed as a possible starting point for future theoretical analyses.